Applications of the Defect of a Finitely Presented Functor

For an abelian category $\mathcal{A}$, the defect sequence $$0⟶F_0⟶F(w(F),\underset{‾}{})⟶F_1⟶0$$ of a finitely presented functor is used to establish the CoYoneda Lemma. An application of this result is the $\mathsf{\text{fp}}$-dual formula which states that for any covariant finitely presented functor $F$, $F^\cong \big(\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} , w(F)\big)$. The defect sequence is shown to be isomorphic to both the double dual sequence $$0⟶{\mathsf{\text{Ext}}}^1(\mathsf{\text{Tr}}F,\mathsf{\text{Hom}})⟶F⟶F^{\ast \ast }⟶{\mathsf{\text{Ext}}}^2(\mathsf{\text{Tr}}F,\mathsf{\text{Hom}})⟶0$$ and the injective stabilization sequence $$0⟶\bar{F}⟶F⟶R^{0}F⟶\tilde{F}⟶0$$ establishing the $\mathsf{\text{fp}}$-injective stabilization formula $\overline{F}\cong $\mathsf{\text{Ext}}$^1($\mathsf{\text{Tr}}$ F,$\mathsf{\text{Hom}}$)$foranyfinitelypresentedfunctor$F$\mathrm{.}Theinjectivesof$$\mathsf{\text{fp}}$($\mathsf{\text{Mod}}$®,$\mathsf{\text{Ab}}$)$ are used to compute the left derived functors $L^k(\hspace{0.05cm}\underline{\
}\hspace{0.1cm} )^$. These functors are shown to detect certain short exact sequences.